STUDENTS PROBABILITY DAY Weizmann Institute of Science March 28, 2007

1. STUDENTS PROBABILITY DAYWeizmann Institute of ScienceMarch 28, 2007 Queueing Networks withInfinite Virtual QueuesAn Example, An Application and a Fundamental Question • Yoni Nazarathy • (Supervisor: Prof. Gideon Weiss) • University of Haifa

2. 6 1/4 1 2 3 3/4 5 4 Multi-Class Queueing Networks (Harrison 1988, Dai 1995,…) Queues Initial Queue Levels Routes Servers Network Dynamics Processing Durations Resource Allocation (Scheduling) Yoni Nazarathy, University of Haifa, 2007

3. m INTRODUCING: Infinite Virtual Queues Regular Queue Infinite Virtual Queue NominalProductionRate m Relative Queue Length: Example Realization Yoni Nazarathy, University of Haifa, 2007

4. 6 1/4 1 2 3 3/4 5 4 MCQN+IVQ Queues Initial Queue Levels Routes Servers Network Dynamics Processing Durations NominalProductionRates Resource Allocation (Scheduling) Yoni Nazarathy, University of Haifa, 2007

5. An Example Yoni Nazarathy, University of Haifa, 2007

6. A Push-Pull Queueing System (Weiss, Kopzon 2002,2006) Server 2 Server 1 PUSH PULL “Inherently Stable” or PULL PUSH “Inherently Unstable” Proportion of time server i allocates to “Pulling” Require Full Utilization Require Rate Stability Fluid Solution: Yoni Nazarathy, University of Haifa, 2007

7. Maximum Pressure (Dai, Lin 2005) • Max-Pressure is a rate stable policy (even when ρ=1). • Push-Pull acts like a ρ=1 System. • As Proven by Dai and Lin, Max-Pressure is rate stable. • But for the Push-Pull system Max-Pressure is not Positive Recurrent: Queue on Server 1 Queue on Server 2 Yoni Nazarathy, University of Haifa, 2007

8. Positive Recurrent Policies Exist!!! Kopzon, Weiss 2006 Kopzon, Weiss 2002 Yoni Nazarathy, University of Haifa, 2007

9. An Application Yoni Nazarathy, University of Haifa, 2007

10. Near Optimal Control over a Finite Time Horizon • Approximation Approach: • 1) Approximate the problem using a fluid system. • 2) Solve the fluid system (SCLP). • 3) Track the fluid solution on-line (Using MCQN+IVQs). • 4) Under proper scaling, the approach is asymptotically optimal. Solution is intractable Finite Horizon Control of MCQN Weiss, Nazarathy 2007 Yoni Nazarathy, University of Haifa, 2007

11. Fluid formulation Server 2 Server 1 3 2 s.t. 1 This is a Separated Continuous Linear Program (SCLP) Yoni Nazarathy, University of Haifa, 2007

12. Fluid solution • SCLP – Bellman, Anderson, Pullan, Weiss. • Piecewise linear solution. • Simplex based algorithm, finds the optimal solution in a finite number of steps (Weiss). • The Optimal Solution: Yoni Nazarathy, University of Haifa, 2007

13. 4 Time Intervals For each time interval, set a MCQN with Infinite Virtual Queues. Yoni Nazarathy, University of Haifa, 2007

14. Now Control the MCQN+IVQ Using a Rate Stable Policy Maximum Pressure (Dai, Lin) is such a policy, even when ρ=1 Yoni Nazarathy, University of Haifa, 2007

15. Example realizations, N={1,10,100} • seed 1 seed 2 seed 3 seed 4 Yoni Nazarathy, University of Haifa, 2007

16. A Fundamental Question Yoni Nazarathy, University of Haifa, 2007

17. Is there a characterization of MCQN+IVQs that allows: • Full Utilization of all the servers that have an IVQ. • Stability of all finite queues. • Proportional equality among production streams. ? Yoni Nazarathy, University of Haifa, 2007

18. ThankYou Yoni Nazarathy, University of Haifa, 2007